Abstract

We prove in this paper that it is much harder to evaluate depth--2, size--N circuits with MOD m gates than with MOD p gates by k--party communication protocols: we show a k--party protocol which communicates O(1) bits to evaluate circuits with MOD p gates, while evaluating circuits with MOD m gates needs\Omega\Gamma N) bits, where p denotes a prime, and m a composite, non-prime power number. As a corollary, for all m, we show a function, computable with a depth--2 circuit with MODm gates, but not with any depth--2 circuit with MOD p gates. Obviously, the k--party protocols are not weaker than the k 0 --party protocols, for k 0 ? k. Our results imply that if there is a prime p between k and k 0 : k ! p k 0 , then there exists a function which can be computed by a k 0 --party protocol with a constant number of communicated bits, while any k--party protocol needs linearly many bits of communication. This result gives a hierarchy theorem for multi--party protocols.

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